## Relations & Functions

# Cartesian Product of Sets

- If A and B are any two non-empty sets, then Cartesian product of A and B is defined as A × B = {(a, b): a ∈ A and b ∈ B}.
- If 'A' has ‘m’ elements and ‘B’ has ‘n’ elements then A × B has mn elements. In general A × B ≠ B × A.

### Part-1: View the Topic in this video From 00:40 To 54:10

### Part-2: View the Topic in this video From 00:40 To 12:48

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1. **Properties of Cartesian Product**

For three sets A, B and C

a) n (A × B) = n (A) n (B)

b) A × B = φ, if either A or B is an empty set.

c) A × (B ∪ C) = (A × B) ∪ (A × C)

d) A × (B ∩ C) = (A × B) ∩ (A × C)

e) A × (B − C) = (A × B) − (A × C)

f) (A × B) ∩ (C × D) = (A ∩ C) × (B ∩ D)

2. A × B = B × A ⇔ A = B

3. a) A × (B*'* ∪ C*'*)*'* = (A × B) ∩ (A × C)

b) A × (B*'* ∩ C*'*)*'* = (A × B) ∪ (A × C)